How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere 28.0 in diameter to produce an electric field of 1500 just outside the surface of the sphere?
1500 whats? Newtons per Coulomb?
Use Gauss' law.
The product of the area (pi D^2/6) and the E-field at the surface is equal to the total charge inside the sphere divided by the constant "epsilon-zero". Find that constant and compute the total charge. Divide that charge by the electron charge (e) to get the number of excess electrons.
It is not necessary that the electrons be distributed uniformly, as long as there is spherical symmetry.
The surface area of a sphere is, of course,
A = 4 pi R^2 = pi D^2,
NOT what I wrote.
I was thinking of the volume, and then got the exponent wrong.
Use the area in Gauss' Law.
To find the number of excess electrons required to produce a certain electric field just outside the surface of a plastic sphere, we can use Coulomb's law and the concept of electric flux.
Here are the step-by-step instructions to calculate the number of excess electrons:
1. Start by determining the charge per electron, denoted as e. The charge of an electron is approximately -1.6 x 10^(-19) coulombs.
2. Calculate the surface area of the plastic sphere using its diameter. The formula for the surface area of a sphere is A = 4πr^2, where A represents the surface area and r is the radius. In this case, since we have the diameter, we can divide it by 2 to obtain the radius.
Given: diameter = 28.0 in
radius = 28.0 in / 2 = 14.0 in
Convert the radius to meters by multiplying by 0.0254 (conversion factor for inches to meters).
radius = 14.0 in * 0.0254 m/in = 0.3556 m
Now calculate the surface area:
A = 4π(0.3556 m)^2
3. Use Gauss's law (which states that the electric flux through a closed surface is proportional to the enclosed charge) to relate the electric field just outside the surface of the sphere to the charge enclosed within the surface. Gauss's law equation is:
Φ = Q / ε₀
Φ represents the electric flux, Q is the charge enclosed within the surface, and ε₀ is the permittivity of free space which is approximately 8.854 x 10^(-12) C^2/(N·m²).
Rearrange the equation to solve for the charge enclosed:
Q = Φ * ε₀
Given: electric field just outside the surface = 1500 N/C
Calculate the electric flux using the formula:
Φ = E * A
Φ = (1500 N/C) * (4π(0.3556 m)^2)
4. Now we have the charge enclosed within the surface. To find the number of excess electrons, divide the charge by the charge per electron (e):
Number of excess electrons = Q / e
Evaluate the expression to get the final answer.
By following these steps, you can calculate the number of excess electrons required to produce the given electric field just outside the surface of an isolated plastic sphere.